Schwarz Lemma application problem

Question

Schwarz lemma says that "Let ${\displaystyle \mathbf {D} =\{z:|z|<1\}}$ be the open unit disk in the complex plane ${\displaystyle \mathbb {C} }$ centered at the origin, and let ${\displaystyle f:\mathbf {D} \rightarrow \mathbb {C} }$ be a holomorphic map such that ${\displaystyle f(0)=0}$ and ${\displaystyle |f(z)|\leq 1}$ on ${\displaystyle \mathbf {D} }$. Then ${\displaystyle |f(z)|\leq |z|}$ for all ${\displaystyle z\in \mathbf {D} }$, and ${\displaystyle |f'(0)|\leq 1}$. Moreover, if ${\displaystyle |f(z)|=|z|}$ for some non-zero ${\displaystyle z}$ or ${\displaystyle |f'(0)|=1}$, then ${\displaystyle f(z)=az}$ for some ${\displaystyle a\in \mathbb {C} }$ with ${\displaystyle |a|=1}$."

I am trying to use this to find an analytic function $f: \mathbb{D} \rightarrow \mathbb{D}$ such that $f(1/8)=4/7$ and $f(4/5)=3/7$? Could someone please explain? Thanks!

Answer 1

You cannot apply the Schwarz lemma directly because $f(0) = 0$ is not given.

But you can use its variant, the Schwarz-Pick theorem: $$ \left|{\frac {f(z_{1})-f(z_{2})}{1-\overline {f(z_{1})}f(z_{2})}}\right|\leq \left|{\frac {z_{1}-z_{2}}{1-\overline {z_{1}}z_{2}}}\right| $$ to show that such a function does not exist.